How about this: Consider the Hilbert space [itex]\mathcal{H}=\ell^2(\mathbb{N})[/itex] of square-summable sequences of reals. Let {en} be the standard o.n. basis for [itex]\mathcal{H}[/itex], and define T on [itex]\mathcal{H}[/itex] by letting T(en)=(1/n)*en and extending linearly. This is a bounded linear operator on [itex]\mathcal{H}[/itex]. Next, consider the space [itex]\mathcal{H} \oplus_2 \mathcal{H}[/itex], which is simply the direct sum of two copies of [itex]\mathcal{H}[/itex] given the 2-norm coordinate wise. (This is still a Hilbert space.) Let A={(x,0) : x in [itex]\mathcal{H}[/itex]} and B={(x,Tx) : x in [itex]\mathcal{H}[/itex]}. Then A and B are subspaces of [itex]\mathcal{H} \oplus_2 \mathcal{H}[/itex], and A+B is closed there iff {Tx : x in [itex]\mathcal{H}[/itex]} is closed in [itex]\mathcal{H}[/itex]. But the range of T is a proper dense subspace of [itex]\mathcal{H}[/itex]. Thus, A+B cannot be closed.